The quasi-strong solutions of a degenerate electrorheological fluid equation

An electrorheological fluid equation with a convection term is considered. Based on the existence of weak solutions, the definition of quasi-strong solution is given and its existence is studied. Different from the other related works, it is permissable that there is a point $(x_0,t_0)\in \Omega\times [0,T)$ such that $a(x_0,t_0)=0$. In consideration of the uniqueness of solution, it is even permissable that the set $\{x\in \Omega: a(x,t)=0\}$ has an interior point, the equation may have hyperbolic characteristic, entropy condition should be quoted. Moreover, the boundary value condition is imposed on a submanifold $\Sigma_1\subset\Omega\times (0,T)$, and the submanifold $\Sigma_1$ can not be expressed as a cylinder such as $\Gamma\times (0,T)$, where $\Gamma\subset\partial\Omega$. Based on this partial boundary value condition, the stability of quasi-strong solutions is established, the uniqueness of entropy solution is proved